Squares of basic geometric shapes table. How to calculate the area of a shape. Triangle. Through base and height
To solve problems in geometry, you need to know formulas - such as the area of a triangle or the area of a parallelogram - as well as the simple tricks that we will talk about.
First, let's learn the formulas for the areas of the figures. We have specially collected them in a convenient table. Print, Learn & Apply!
Of course, not all geometry formulas are in our table. For example, to solve problems in geometry and stereometry in the second part of the profile USE in mathematics, other formulas for the area of a triangle are also used. We will definitely tell you about them.
But what if you need to find not the area of a trapezoid or triangle, but the area of some complex figure? There are universal ways! Let's show them with examples from the FIPI job bank.
1. How to find the area of a non-standard shape? For example, an arbitrary quadrilateral? A simple trick is to break this figure into those that we all know about, and find its area - as the sum of the areas of these figures.
Divide this quadrilateral with a horizontal line into two triangles with a common base equal to. The heights of these triangles are 
and . Then the area of the quadrilateral is equal to the sum of the areas of two triangles:.
Answer: .
2. In some cases, the area of a figure can be represented as the difference between some areas.
It is not so easy to calculate what the base and height in this triangle are equal to! But we can say that its area is equal to the difference between the areas of a square with a side and three right-angled triangles... Do you see them in the picture? We get:.
Answer: .
3. Sometimes in the task it is necessary to find the area not of the whole figure, but of its part. Usually we are talking about the area of a sector - a part of a circle. Find the area of a sector of a circle of radius, the length of an arc of which is .
In this picture, we see part of a circle. The area of the whole circle is equal since. It remains to find out which part of the circle is depicted. Since the length of the entire circle is (since), and the length of the arc of this sector is , therefore, the length of the arc is one-fold less than the length of the entire circle. The angle at which this arc rests is also one-fold less than a full circle (that is, degrees). This means that the area of the sector will be one times less than the area of the entire circle.
Area of a geometric figure- a numerical characteristic of a geometric figure showing the size of this figure (part of the surface bounded by the closed contour of this figure). The size of the area is expressed by the number of square units contained in it.
Area formulas for a triangle
- Formula for the area of a triangle by side and height
Area of a triangle equal to half the product of the length of the side of the triangle by the length of the height drawn to this side - The formula for the area of a triangle on three sides and the radius of the circumscribed circle
- The formula for the area of a triangle on three sides and the radius of the inscribed circle
Area of a triangle is equal to the product of the half-perimeter of the triangle and the radius of the inscribed circle. where S is the area of the triangle,
- the lengths of the sides of the triangle,
- the height of the triangle,
- the angle between the sides and,
- radius of the inscribed circle,
R is the radius of the circumscribed circle,
Area of a square formulas
- Formula for the area of a square by the length of a side
Square area is equal to the square of the length of its side. - Formula for the area of a square by the length of the diagonal
Square area is equal to half the square of the length of its diagonal.S = 1 2 2 where S is the area of the square,
- the length of the side of the square,
- the length of the diagonal of the square.
Formula for the area of a rectangle
- Rectangle area equal to the product of the lengths of its two adjacent sides
where S is the area of the rectangle,
- the lengths of the sides of the rectangle.
Parallelogram area formulas
- Formula for the area of a parallelogram in terms of side length and height
Parallelogram area - Formula for the area of a parallelogram on two sides and the angle between them
Parallelogram area equal to the product of the lengths of its sides multiplied by the sine of the angle between them.a b sin α
where S is the area of the parallelogram,
- the lengths of the sides of the parallelogram,
- length of parallelogram height,
- the angle between the sides of the parallelogram.
Rhombus area formulas
- Formula for the area of a rhombus by side length and height
Rhombus area is equal to the product of the length of its side and the length of the height lowered to this side. - Formula for the area of a rhombus by side length and angle
Rhombus area is equal to the product of the square of the length of its side and the sine of the angle between the sides of the rhombus. - Formula for the area of a rhombus by the lengths of its diagonals
Rhombus area is equal to half the product of the lengths of its diagonals. where S is the area of the rhombus,
- the length of the rhombus side,
- the length of the height of the rhombus,
- the angle between the sides of the rhombus,
1, 2 - the lengths of the diagonals.
Area formulas for a trapezoid
- Heron's formula for trapezoid
Where S is the area of the trapezoid,
- the length of the bases of the trapezoid,
- the length of the lateral sides of the trapezoid,
Area formula is necessary to determine the area of a figure, which is a real-valued function defined on a certain class of figures in the Euclidean plane and satisfying 4 conditions:
- Positiveness - The area cannot be less than zero;
- Normalization - a square with a side of one has an area of 1;
- Congruence - Congruent shapes have equal area;
- Additivity - the area of the union of 2 figures without common internal points is equal to the sum of the areas of these figures.
| Geometric figure | Formula | Drawing |
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The result of the addition of the distances between the midpoints of opposite sides of a convex quadrilateral will be equal to its semiperimeter. |
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Circle sector. The area of a sector of a circle is equal to the product of its arc and half the radius. |
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Segment of a circle. To obtain the area of the ASB segment, it is sufficient to subtract the area of the triangle AOB from the area of the AOB sector. |
S = 1/2 R (s - AC) |
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The area of an ellipse is equal to the product of the lengths of the major and minor semiaxes of the ellipse by the number pi. |
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Ellipse. Another option for calculating the area of an ellipse is through its two radii. |
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Triangle. Through the base and height. Formula for the area of a circle in terms of its radius and diameter. |
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Square . Through his side. The area of a square is equal to the square of the length of its side. |
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Square. Through its diagonals. The area of a square is half the square of the length of its diagonal. |
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Regular polygon. To determine the area of a regular polygon, it is necessary to split it into equal triangles, which would have a common vertex in the center of the inscribed circle. |
S = r p = 1/2 r n a |
The knowledge of how to measure the Earth dates back to antiquity and gradually evolved into the science of geometry. This word is translated from the Greek language - "surveying".
The measure of the length and width of a flat area of the Earth is area. In mathematics, it is usually denoted by the Latin letter S (from the English "square" - "area", "square") or Greek letterσ (sigma). S denotes the area of a figure on a plane or the surface area of a body, and σ is the cross-sectional area of a wire in physics. These are the main symbols, although there may be others, for example, in the field of strength of materials, A is the cross-sectional area of the profile.
In contact with
Calculation formulas
Knowing the areas of simple shapes, you can find the parameters of more complex... Ancient mathematicians developed formulas by which they can be easily calculated. Such figures are a triangle, quadrilateral, polygon, circle.
To find the area of a complex planar figure, it is broken down into many simple figures, such as triangles, trapezoids, or rectangles. Then, by mathematical methods, a formula for the area of this figure is derived. A similar method is used not only in geometry, but also in mathematical analysis to calculate the areas of figures bounded by curves.
Triangle
Let's start with the simplest shape - a triangle. They are rectangular, isosceles and equilateral. Take any triangle ABC with sides AB = a, BC = b and AC = c (∆ ABC). To find its area, let us recall the theorems of sines and cosines known from the school mathematics course. Releasing all calculations, we come to the following formulas:
- S = √ is the well-known Heron formula, where p = (a + b + c) / 2 is the half-perimeter of a triangle;
- S = a h / 2, where h is the height lowered to side a;
- S = a b (sin γ) / 2, where γ is the angle between sides a and b;
- S = a b / 2, if ∆ ABC is rectangular (here a and b are legs);
- S = b² (sin (2 β)) / 2, if ∆ ABC is isosceles (here b is one of the “hips”, β is the angle between the “hips” of the triangle);
- S = a² √¾ if ∆ ABC is equilateral (here a is the side of the triangle).
Quadrilateral
Let there be a quadrilateral ABCD with AB = a, BC = b, CD = c, AD = d. To find the area S of an arbitrary 4-gon, you need to divide it by the diagonal into two triangles, the areas of which S1 and S2 are generally not equal.
Then, using the formulas, calculate them and add them, that is, S = S1 + S2. However, if a 4-gon belongs to a certain class, then its area can be found using the previously known formulas:
- S = (a + c) h / 2 = e h if the 4-gon is a trapezoid (here a and c are bases, e is middle line trapezium, h - height lowered to one of the bases of the trapezoid;
- S = a h = a b sin φ = d1 d2 (sin φ) / 2, if ABCD is a parallelogram (here φ is the angle between sides a and b, h is the height dropped to side a, d1 and d2 are diagonals);
- S = a b = d² / 2, if ABCD is a rectangle (d is a diagonal);
- S = a² sin φ = P² (sin φ) / 16 = d1 d2 / 2, if ABCD is a rhombus (a is the side of the rhombus, φ is one of its corners, P is the perimeter);
- S = a² = P² / 16 = d² / 2 if ABCD is a square.
Polygon
To find the area of an n-gon, mathematicians split it into the simplest equal figures-triangles, find the area of each of them, and then add them. But if the polygon belongs to the class of regular ones, then use the formula:
S = anh / 2 = a² n / = P² /, where n is the number of vertices (or sides) of the polygon, a is the side of an n-gon, P is its perimeter, h is an apothem, that is, a segment drawn from the center of the polygon to one of its sides at an angle of 90 °.
A circle
A circle is a perfect polygon with an infinite number of sides.... We need to calculate the limit of the expression on the right in the formula for the area of a polygon with the number of sides n tending to infinity. In this case, the perimeter of the polygon will turn into the circumference of a circle of radius R, which will be the boundary of our circle, and will become equal to P = 2 π R. Substitute this expression into the above formula. We'll get:
S = (π² R² cos (180 ° / n)) / (n sin (180 ° / n)).
Let us find the limit of this expression as n → ∞. To do this, take into account that lim (cos (180 ° / n)) as n → ∞ is equal to cos 0 ° = 1 (lim is the limit sign), and lim = lim as n → ∞ is equal to 1 / π (we translated the degree measure to radian using the ratio π rad = 180 °, and applied the first remarkable limit lim (sin x) / x = 1 as x → ∞). Substituting the obtained values into the last expression for S, we arrive at the well-known formula:
S = π² R² 1 (1 / π) = π R².
Units
System and non-system units are used... System units refer to SI (International System). It is a square meter (square meter, m²) and units derived from it: mm², cm², km².
In square millimeters (mm²), for example, they measure the cross-sectional area of wires in electrical engineering, in square centimeters (cm²) - cross-sections of a beam in structural mechanics, in square meters (m²) - apartments or houses, in square kilometers (km²) - territories in geography ...
However, sometimes non-systemic units of measurement are also used, such as: weaving, ar (a), hectare (ha) and acre (ac). Here are the following relationships:
- 1 hundred square meters = 1 a = 100 m² = 0.01 hectares;
- 1 hectare = 100 a = 100 ares = 10000 m² = 0.01 km² = 2.471 ac;
- 1 ac = 4046.856 m2 = 40.47 a = 40.47 ares = 0.405 hectares.
